


def 		_A		(		_lowercase )					-> None:


     """simple docstring"""


     __UpperCamelCase =			generate_pascal_triangle(_lowercase )
     for row_idx in range(_lowercase ):
          # Print left spaces
          for _ in range(num_rows - row_idx - 1 ):
               print(end=' ' )
          # Print row values
          for col_idx in range(row_idx + 1 ):
               if col_idx != row_idx:
                    print(triangle[row_idx][col_idx]					,  end=' ' )
               else:
                    print(triangle[row_idx][col_idx]					,  end='' )
          print()






def 		_A		(		_lowercase )					-> list[list[int]]:


     """simple docstring"""


     if not isinstance(_lowercase					,  _lowercase ):
          raise TypeError('The input value of \'num_rows\' should be \'int\'' )

     if num_rows == 0:
          return []
     elif num_rows < 0:
          raise ValueError(
              'The input value of \'num_rows\' should be greater than or equal to 0' )

     __UpperCamelCase =			[]
     for current_row_idx in range(_lowercase ):
          __UpperCamelCase =			populate_current_row(_lowercase					,  _lowercase )
          triangle.append(_lowercase )
     return triangle






def 		_A		(		_lowercase					,  _lowercase )					-> list[int]:


     """simple docstring"""


     __UpperCamelCase =			[-1] * (current_row_idx + 1)
     # first and last elements of current row are equal to 1
     __UpperCamelCase,    __UpperCamelCase =			1, 1
     for current_col_idx in range(1					,  _lowercase ):
          calculate_current_element(
              _lowercase					,  _lowercase					,  _lowercase					,  _lowercase )
     return current_row






def 		_A		(		_lowercase					,  _lowercase					,  _lowercase					,  _lowercase					,  )					-> None:


     """simple docstring"""


     __UpperCamelCase =			triangle[current_row_idx - 1][current_col_idx - 1]
     __UpperCamelCase =			triangle[current_row_idx - 1][current_col_idx]
     __UpperCamelCase =			above_to_left_elt + above_to_right_elt






def 		_A		(		_lowercase )					-> list[list[int]]:


     """simple docstring"""


     if not isinstance(_lowercase					,  _lowercase ):
          raise TypeError('The input value of \'num_rows\' should be \'int\'' )

     if num_rows == 0:
          return []
     elif num_rows < 0:
          raise ValueError(
              'The input value of \'num_rows\' should be greater than or equal to 0' )

     __UpperCamelCase =			[[1]]

     for row_index in range(1					,  _lowercase ):
          __UpperCamelCase =			[0] + result[-1] + [0]
          __UpperCamelCase =			row_index + 1
          # Calculate the number of distinct elements in a row
          __UpperCamelCase =			sum(divmod(_lowercase					,  2 ) )
          __UpperCamelCase =			[
              temp_row[i - 1] + temp_row[i] for i in range(1					,  distinct_elements + 1 )
          ]
          __UpperCamelCase =			row_first_half[: (row_index + 1) // 2]
          row_second_half.reverse()
          __UpperCamelCase =			row_first_half + row_second_half
          result.append(_lowercase )

     return result






def 		_A		(		)					-> None:


     """simple docstring"""


     from collections.abc import Callable
     from timeit import timeit

     def benchmark_a_function(_lowercase					,  _lowercase ) -> None:
          __UpperCamelCase =			f'''{func.__name__}({value})'''
          __UpperCamelCase =			timeit(f'''__main__.{call}'''					,  setup='import __main__' )
          # print(f"{call:38} = {func(value)} -- {timing:.4f} seconds")
          print(f'''{call:38} -- {timing:.4f} seconds''' )

     for value in range(15 ):  # (1, 7, 14):
          for func in (generate_pascal_triangle, generate_pascal_triangle_optimized):
               benchmark_a_function(_lowercase					,  _lowercase )
          print()


if __name__ == "__main__":
    import doctest

    doctest.testmod()
    benchmark()







'''simple docstring'''



from collections import defaultdict
def        UpperCAmelCase (    A	:					int					):
							SCREAMING_SNAKE_CASE       :   List[Any]         =      1
							SCREAMING_SNAKE_CASE       :   Dict         =      True
							for v in tree[start]:
														if v not in visited:
																					ret += dfs(A					)
							if ret % 2 == 0:
														cuts.append(A					)
							return ret
def        UpperCAmelCase (    ):
							dfs(1					)


if __name__ == "__main__":
							lowerCAmelCase_      ,				lowerCAmelCase_  :							Dict             =    10, 9
							lowerCAmelCase_  :							Dict             =    defaultdict(list)
							lowerCAmelCase_  :							dict[int, bool]             =    {}
							lowerCAmelCase_  :							list[int]             =    []
							lowerCAmelCase_  :							Optional[Any]             =    0
							lowerCAmelCase_  :							Dict             =    [(2, 1), (3, 1), (4, 3), (5, 2), (6, 1), (7, 2), (8, 6), (9, 8), (10, 8)]
							for u, v in edges:
														tree[u].append(v)
														tree[v].append(u)
							even_tree()
							print(len(cuts) - 1)




